Non-Invertible Defects in Nonlinear Sigma Models and Coupling to Topological Orders

TL;DR

The paper develops a comprehensive framework for defects in nonlinear sigma models, distinguishing electric defects arising from topological actions on submanifolds and magnetic defects defined by boundary conditions tied to the target space topology. It shows that in the presence of topological terms, magnetic defects can become non-invertible and can fuse and braid nontrivially, with their interactions giving rise to higher-group structures linking defects of different dimensions. The authors extend this by coupling sigma models to TQFTs, showing how defect attachment modifies fusion and braiding, and they illustrate these ideas across examples including toroidal target spaces, axion-monopole systems, and axion gauge theories in 3+1D. The work further connects these defect structures to energy-scale constraints, symmetry breaking phases in gauge theories, and concrete UV–IR symmetry matchings in QED/QCD-like settings, providing a robust toolkit for analyzing non-invertible topological phenomena in high-dimensional quantum field theories.

Abstract

Nonlinear sigma models appear in a wide variety of physics contexts, such as the long-range order with spontaneously broken continuous global symmetries. There are also large classes of quantum criticality admit sigma model descriptions in their phase diagrams without known ultraviolet complete quantum field theory descriptions. We investigate defects in general nonlinear sigma models in any spacetime dimensions, which include the "electric" defects that are characterized by topological interactions on the defects, and the "magnetic" defects that are characterized by the isometries and homotopy groups. We use an analogue of the charge-flux attachment to show that the magnetic defects are in general non-invertible, and the electric and magnetic defects form junctions that combine defects of different dimensions into analogues of higher-group symmetry. We explore generalizations that couple nonlinear sigma models to topological quantum field theories by defect attachment, which modifies the non-invertible fusion and braiding of the defects. We discuss several applications, including constraints on energy scales and scenarios of low energy dynamics with spontaneous symmetry breaking in gauge theories, and axion gauge theories.

Figures (3)

• Figure 1: There are magnetic defects of codimension $(k-1)$ that are the analogues of "Dirac strings" for improperly quantized magnetic defects of codimension $k$, for very short "Dirac string" this is a properly quantized magnetic defect of codimension $k$.
• Figure 2: Magnetic defect of codimension $k$ is attached to electric defect of codimension $(k-1)$ given by the integration of the topological interaction $\omega^{(D)}$ over the $S^{k-1}$ fiber with background configuration specified by the magnetic defect $m_k: S^{k-1}\rightarrow M$ for the target space $M$. The emitted electric defect is computed by the product $i_{m_k}\omega^{(D)}$.
• Figure 3: In the presence of topological action, the trivalent junction of magnetic defects emits an electric defect.